Method of measuring factor of stress concentration by utilizing ultrasound

ABSTRACT

The present invention provides a method of measuring, by utilizing ultrasound, the factor of stress concentration at a stress-concentrated portion of a member, comprising the steps of emitting ultrasound for incidence upon the stress-concentrated portion of the member in stressed state, increasing the stress in said stressed state, comparing the acoustic pressure of the reflected wave from said stress-concentrated portion between before and after the stress is changed, for thereby measuring the factor of stress concentration and by which the factor of stress concentration of a member composing a machine or the like in a static or dynamic state can be measured easily, real-time, quantitatively, nondestructively and highly accurately.

BACKGROUND OF THE INVENTION

(a) Field of the Invention

The present invention relates to a method of measuring, by utilizingultrasound, the factor of stress concentration at the portion of amechanical member, structural member or the like where stress isconcentrated (will be referred to as "stress-concentrated portion"hereinafter).

The stress-concentrated portions of members, to which the presentinvention is applicable, include structural notched or cut portions suchas hole, key way formed in members composing, for example, machineries,structures, etc. in all fields of industry, as well as undercut, blowhole, nest, crack, etc. developed in the manufacturing processes such aswelding, forging, molding, etc. Also, they include the portions ofmaterials, like flaw, crack, etc., of which the section varies abruptly.

More particularly, the present invention concerns a method of measuringthe factor of stress concentration at the stress-concentrated portion ofa member made of a metal or nonmetal (glass, ceramic, synthetic resin,etc.) and through which ultrasound can be propagated.

(b) Description of the prior art

Analysis concerning the stress-concentrated portion of a member, andsetting of a factor of stress concentration at such portion, amongothers, are essential in designing and manufacturing a machine orstructure in order to prevent any breakdown or damage of them and alsoto improve the safety and reliability. In the field of technology towhich the present invention belongs, however, the method of measuringthe factor of stress concentration of a real object easily, real-timeand quantitatively is very important and necessary, but has not yet beenestablished, for analysis of the stress concentration at the portionsjoined by the welding having been utilized from the old time and ofwhich the application is very wide, and even for analysis of the stressconcentration at the structural notched or cut portions of mechanicalparts, typically, hole or key way formed by utilizing no welding and ofwhich, it is said, the study has been highly advanced. It is very hardto theoretically analyze the mechanism of stress concentration even inthese fields.

To solve the above problem, various methods of measuring the factor ofstress concentration have been proposed. However, they include only theexperimental methods of measuring using the photo-elasticity. That is,model experiments were made for analysis and study of only the typicalmodels such as those in which circular holes or elliptical cavities weredeveloped inside or on the surface of an elastic object like steel or inwhich U- or V-shaped notches were found on the surface of such object.These proposed methods were static, qualitative and indirect ones, andthey were reported by H. Neuber and R. B. Heywood in 1958.

However, since the notches developed in the surface of bead weld vary inshape from one to another depending upon the method and kind of weldingand also the shape of welded joint, the reproducibility can hardly beexpected of the conventional method of measuring the factor of stressconcentration at the weld zone. Therefore, it is practically impossibleto condition any test piece of a weld zone into a predetermined model.In measurement of the factor of stress concentration at the weldedportion, even the method of measuring by using the photo-elasticity,namely, a static, qualitative and indirect one, has some problems inapplications and it is difficult to employ the method, as will beexplained below.

(1) Although it is necessary to prepare a model as test piece, thismodeling is impossible in practice for the above reasons.

(2) Even if such model could be prepared, since it is to be made from amaterial such as high-molecular epoxy resin, diarylphthalate resin orthe like, differences in material, dimensions, working precision, etc.from the actual test piece are inevitably encountered which will alsoaffect the fringe order of the stress-concentrated portion, resulting inno stable fringe order.

(3) Because of the material properties of the model, it is extremelydifficult to make any acute-angle notch and so the reproduction of anystress concentration which will occur in the actual object is notexpectable.

As a method of measuring the factor of stress concentration using nophoto-elasticity, use of an electric resistance strain gauge (will bereferred to as "strain gauge" hereinafter) is known; however, the straingauge cannot be attached in any acute-angle notch. Even if the notch hasan area wide enough to receive the strain gauge, the attaching of thestrain gauge will cause of the stress-concentrated portion to have theproperties changed. Thus, this method is also disadvantageous inimpossibility of measuring any real factor of stress concentration.Namely, it is just an experimental method of measuring and has not beenprevailing.

The present invention primarily seeks to provide a method of measuring,by utilizing ultrasound, the factor of stress concentration at thestress-concentrated portion of a mechanical or structural member, bywhich anyone can real-time measure the factor of stress concentrationeasily, quantitatively and with a high accuracy without changing thestate in which the stress continuously acts on the stress-concentratedportion (will be referred to as "stressed state" hereinafter) and thenature of the portion in the stressed state.

Also the present invention seeks to provide a method of measuring thefactor of stress concentration, which can measure the factor of a staticstress concentration for an extremely short time (a few seconds) andalso the factor of a dynamic stress concentration directly.

Furthermore, the present invention seeks to provide a method ofmeasuring the factor of stress concentration, which can always providefor a highly accurate, quantitative measurement without being influencedby the shape and roughness of the surface on which the ultrasound probeis placed and even with more or less difference in placement of theprobe.

DISCLOSURE OF THE INVENTION

The above objects are attainable by providing a method of measuring,according to the present invention, the factor of stress concentrationat the stress-concentrated portion of a member in the stressed state,comprising the steps of emitting ultrasound for incidence upon saidstress-concentrated portion of the member in said stressed state;increasing or decreasing the stress in said stressed state; emittingultrasound again for incidence upon said stress-concentrated portion inthe state of said increased or decreased stress; comparing the acousticpressure of the former reflected ultrasonic wave with that of the latterone also from said stress-concentrated portion, for thereby measuringthe factor of stress concentration at the stress-concentrated portion ofsaid member taking as evaluation index the change ratio of the acousticpressure of the reflected ultrasonic wave.

This characteristic of the present invention is to utilize the fact thatwhen a member to be measured is continuously stressed and ultrasound isemitted for incidence upon the stress-concentrated portion of themember, there is a correlation between the change in acoustic pressureof the reflected ultrasonic wave from said stress-concentrated portionand the stress working on the stress-concentrated portion (will bereferred to as "working stress" hereinafter).

FIGS. 1 to 3 explain the basic principle of the inventive method ofmeasuring the factor of stress concentration. In these Figures, thereference numeral 1 indicates a member to be measured, 2 astress-concentrated portion of the member 1, 3 a probe secured by anadhesive on the surface of the member 1 and which emits an ultrasoundfor incidence toward the stress-concentrated portion 2, 4 an incidentwave of the ultrasound P_(O) acoustic pressure of the incident wave, and5 a reflection, or reflected wave, of the incident wave 4 from thestress-concentrated portion 2, this reflected wave being received by theprobe 3.

When a tensile force F₁ acts on the to-be-measured member 1 within thelimits of elasticity, a tensile stress develops in the member 1, a meanstress σ_(n1) obtained by dividing the tensile force F₁ by a minimumsectional area, namely, an area except for the stress-concentratedportion 2, of the to-be-measured member 1 is distributed, and a maximumlocal stress σ_(max) corresponding to the tensile stress F₁ develops inthe notched bottom of the stress-concentrated portion 2. By emitting anultrasound under an acoustic pressure P₀ from the probe 3 for incidencetoward the stress-concentrated zone 2 in the above condition, theincident wave 4 reaches the stress-concentrated portion 2, reflectedthere and received as a reflected wave 5 of the acoustic pressure P₁ bythe probe 3. Next, by increasing the tensile force F₁ up to F₂, the meanstress σ_(n1) increases up to a mean stress σ_(n2). The result of amicroscopic observation of the change in shape of the notched bottom aof the stress-concentrated portion 2 due to the increase in means stressfrom σ_(n1) to σ_(n2) is shown in FIGS. 2 and 3. In FIG. 2, the meanstress is σ_(n1), the notched bottom retains its initial shape and theacoustic pressures of the incident wave 4 and reflected wave 5,respectively, are P₀ and P₁, respectively. FIG. 3 shows a case in whichthe mean stress is increased to σ_(n2) with the notched bottomelastoplastically changed in shape, showing the changes in shape, area,etc. which influence the acoustic pressure of the reflected wave 5. Theincident wave 4 emitted under the acoustic pressure P₀ is so influencedby the plasto-elastical deformation of the notched bottom of thestress-concentrated portion 2 as to have the acoustic pressuredecreased, and thus the incident wave becomes a reflected wave 5 of anacoustic pressure P₂ which is to be received by the probe 3. As the meanstress changes, the acoustic pressure changes correspondingly in thismanner. However, even if the mean stress remains constant, not changed,the reflected wave has the acoustic pressure changed as the case may be.This is because the notch in the stress-concentrated portion 2 is sharp,so that the notched bottom is deformed elastoplastically in the samemanner as if the working stress increases although it does not actuallyincrease, resulting in the change of the acoustic pressure of thereflected wave as influenced by the elasto-plastical deformation. Thatis to say, the greater the factor of stress concentration, the largerthe elastoplastical deformation of the notched bottom of thestress-concentrated portion 2 and also the change in acoustic pressureof the reflected wave. The inventive method of measuring is such thatwhen a stress works continuously on the member to be measured, thefactor of stress concentration is measured from the rate of change inacoustic pressure of the reflected ultrasonic wave caused by theelastoplastical deformation at the stress-concentrated portion.

The inventive method of measuring the factor of stress concentrationwill be further explained using the mathematical expressions. The ratioP₁ /P₀ between the acoustic pressure P₁ of reflected wave 5 and that P₀of incident wave 4 in case of the mean stress of member to be measured 1being σ_(n1), and the ratio P₂ /P₀ between the acoustic pressure P₂ ofreflected wave 5 and that P₀ of incident wave 4 in case of the meanstress being σ_(n2) has the values given by the following expressions(1) and (2), respectively:

    P.sub.1 /P.sub.0 =C.sub.1 /f(σ.sub.n1)               (1)

    P.sub.2 /P.sub.0 =C.sub.2 /f(σ.sub.n2)               (2)

where

f(σ_(n1)), f(σ_(n2)): Functions of mean stresses σ_(n1) and σ_(n2),respectively; these functions have larger values along with the increaseof the mean stresses σ_(n1) and σ_(n2).

C1, C2: Constants of proportion, each determined depending upon thedimensions, shape, etc. of the stress-concentrated portion 2

The ratio between the expressions (1) and (2), namely, the ratio betweenP₁ /P₂ and P₂ /P₀, is as follows: ##EQU1## where C3: Constant ofproportion (=C₁ /C₂)

P₁ /P₂ : Function of mean stress (which is apparent from the expression(3))

Also, the factor of stress concentration has the value given by theexpression as follows: ##EQU2## where σ_(max) : Maximum stress developedin the stress-concentrated portion 2

σ_(n) : Mean stress

The expression (3) can be rewritten as follows because of the expression(3): ##EQU3## where C4: Constant of proportion

f(α): Function of factor of stress concentration α

The acoustic pressure P₁ of reflected wave in case the mean stress isσ_(n1) can be expressed as an acoustic pressure for a small reflector asin the following:

    P.sub.1 =C.sub.5 ·D.sup.2 (β,φ)·G.sup.2 (X)·a.sub.1 /S·r.sub.1 ·CR.sub.1 ·e.sup.-2α.sbsp.0.sup.X                    ( 6)

Also, the acoustic pressure P₂ of reflected wave in case the mean stressis σ_(n2) is expressed as follows:

    P.sub.2 =C.sub.6 ·D.sup.2 (β,φ)·G.sup.2 (X)·a.sub.2 /S·r.sub.2 ·CR.sub.2 ·e.sup.-2α.sbsp.0.sup.X                    ( 7)

where

D(β,φ): Directivity of ultrasound

φ: Azimuthal angle of ultrasound

β: Directional angle of ultrasound

G(X): Range characteristic of the acoustic pressure on acoustic axis(beam axis) of ultrasound

a₁, a₂ : Area in reflective zone, which has an influence on the acousticpressure of reflected wave

S: Apparent area of vibrator

CR₁, CR₂ : Shape factors of reflective zone

α₀ : Attenuation factor of ultrasound

r₁, r₂ : Reflection coefficient of ultrasound

X: Path length of ultrasound beam from incidence point

C₅, C₆ : Constants of proportion

The ratio between the expressions (6) and (7), namely, the ratio P₁ /P₂,can be expressed as follows: ##EQU4## The following expression (9) isobtainable from the expressions (5) and (8): ##EQU5##

It is apparent from the expression (9) that the acoustic pression ratioP₁ /P₂ when the working stress is changed is a function of the factor ofstress concentration α and also a function of the change inacoustic-pressure reflection factor ratio due to any microscopicalelastoplastic deformation in the stress-concentrated portion as theworking stress changes. The expression (9) explains the basic principleof the present invention.

As having been described in the foregoing, the inventive method ofmeasuring the factor of stress concentration is to measure a change ofmean stress σ_(n) and a ratio between different reflected-wave acousticpressures P₁ and P₂ in the state with the mean-stress change, forthereby determining a factor of stress concentration from the expression(9). These values can be easily determined by means of a well-knownmeasuring apparatus and calculator. As apparent from the expression (3),the value of incident-wave acoustic pressure P₀ is set off, so that themeasurement is not affected by the acoustic pressure P₀ of the incidentwave, for example, by the contact of the probe, shape and roughness of aportion to which the probe is applied. Thus, a high accuracy ofmeasurement can be attained. Therefore, the inventive method ofmeasuring the factor of stress concentration permits to measure thefactor of a static stress concentration as well as of a dynamic stressconcentration in a real object quantitatively, highly accurately andreal-time.

The basic principle of the inventive method was proved by theexperiments as follows:

It will be explained with reference to FIGS. 4 to 15. A portion of thetest piece used in the experiments is shown in FIG. 4. The test piecehas the dimensions of 35 mm (in large width B) or 20 mm (in small widthb), 200 mm (in length l₁ and l₂ with the large and small widths) and 50mm (in constant thickness t); the radii of curvature ρ of the filletweld in which stress is concentrated are in 7 kinds: 0.08, 0.21, 0.39,0.66, 1.42, 1.90 and 2.75 mm. For facilitating to measure the radius ofcurvature ρ, a model was made of the fillet weld by means of a modelduplicator, cut into slices, and each slice was put in a light projectorand magnified more than 20 times for measurement. The test piece wasshaped by modeling the fillet weld zone of a Tee weld joint as in FIG.5a with the radius of curvature ρ as shown in FIG. 5b and machining thefillet weld zone. The radii of curvature ρ are in 7 kinds as as in theabove. The material of the test pieces was SM50A (JIS G3106 "Rolledsteel for welded structure"), and the accuracy of finishing was 240 μmRzat the portion of the radius of curvature ρ and 120 μmRz at the otherportions (JIS B0601 "Mean roughness at cross point in definition anddisplay of surface roughness).

The experiments were conducted using the apparatus shown in FIG. 6.First, the probe 3 placed on the surface of the test piece 6 was securedon the test-piece surface in such a manner that the ultrasonic waveemitted from the probe 3 was incident toward the stress-concentratedportion 2. The probe 3 was connected to an ultrasonic flaw detector 7 bypulse echo technique (will be referred to as "ultrasonic flaw detector"hereinafter) widely used as nondestructive flaw detector for metals.When the test piece 6 mounted in place on the Amsler universal testingmachine (not shown) was applied with a tensile load F in the directionof arrow with the load changed a reflected acoustic pressure developedin the stress-concentrated portion 2 correspondingly to each ofdifferent mean stresses ρ_(n) working on each of the tensile loads F wasdisplayed on a CRT display 8 of the ultrasonic flaw detector 7. Theacoustic pressure was read for measurement. The above-mentioned meanstress ρ_(n) is a value obtained by dividing the tensile load F by thearea of a parallel zone with the small width b. The ultrasonic flawdetector 7 was SM-80, SM-90, etc. made by Tokyo Keiki Co., Ltd. and theprobe 3 was an angle beam probe made by Japan Probe Co., Ltd. The probes3 used in the experiments were in 5 kinds: 2Z10×10A70, 5Z10×10A70,5Z10×10A60 5Z10×10A45 and 7Z10×10A70. Since the change in acousticpressure of the reflected wave, having taken place in thestress-concentrated portion 2 in response to the change of mean stressσ_(n) was very small, the acoustic pressure of reflected wave wasmeasured by the following method for facilitating the read andmeasurement:

(1) With the rejection control turned off, the echo at a lower levelthan a predetermined level on the CRT display 8 is not inhibited.

(2) A section paper was attached on the scale panel on the CRT display 8and the portion equal to 10% of the scale panel was regularly divided by20; namely, one division was made 0.5% of the scale panel.

(3) The reflected-wave acoustic pressure from the stress-concentratedportion 2 where the mean stress was σ_(n) was zero set at the center(equal to 50%) of the scale panel on the CRT display 8.

(4) As the test piece 6 was applied with various tensile loads F, thereflected-wave acoustic pressure from the stress-concentrated portion 2was read by the unit of 0.5%/ division at each of the tensile load F.

(5) The value of the acoustic pressure read by the unit of 0.5%/divisionat the step (4) above was displayed as echo height having been convertedinto decibel value, not acoustic pressure.

The reflected wave derived from the incident wave 4 emitted forincidence upon the stress-concentrated portion 2 is displayed on the CRTdisplay 8 taking as horizontal axis the distance x from the incidencepoint of the incident wave 4 to the stress-concentrated portion 2 asshown in FIG. 6 and as the vertical axis the height h of the echo 10displayed in dB. The reference numeral 9 indicates an echo of thetransmitted ultrasonic wave. This is illustrated in FIG. 7.

The experiments were also conducted on the test pieces with the radii ofcurvature ρ of the fillet weld zone being in 7 kinds as having beendescribed. Furthermore, the experiments were done with the differentradii of curvature ρ and small width b in combination with said 5 typesof probes. As the results, the basic principle of the inventive methodof measuring the factor of stress concentration was proved as follows:

(1) The result of the experiment conducted on the relation between thechange of means stress σ_(n) and echo height h is shown in FIG. 8. Thetest piece used in this experiment has the same shape as in FIG. 4 andthe dimensions: B=25. b=20, l₁ =l₂ =200 and t=50 (all in mm) which wereconstant. The radii of curvature ρ of the fillet weld zone were in 7kinds: 0.08, 0.21, 0.39, 0.66, 1.41, 1.90 and 2.75 (all in mm). Theprobe used was 5Z10×10A70 of which the frequency was 5 MHz. Theexperiment was conducted in the manner as having been described withreference to FIG. 6 in the foregoing. As seen from FIG. 8, when the meanstress σ_(n) is on the order of 10 to 15 kg/cm², all the echo heights hdisplayed in db decrease almost linearly whether the radius of curvatureρ is large or small, so that the linear expression (10) is established:

    h=-aσ.sub.n                                          ( 10)

where

a: Constant of proportion indicative of gradient

Next, while the means stess σ_(n) is being 10 to 20 kg/mm², the echoheight h is nearly constant although the radius of curvature ρ varies alittle from one to another. When the mean stress σ_(n) exceeds theabove-mentioned range, the echo height h increases linearly with thetest pieces of relatively large radii of curvature ρ among the 7 kindsof test pieces, and the following expression (11) is established:

    h=bσ.sub.n +C                                        (11)

where

b: Constant of proportion

C: Constant

Within the stress region where the relation (11) is established, thestress-concentrated portion is within the limits of elasticity, and themaximum stress σ_(max) does not exceed the yield point σ_(y) of thetest-piece material; therefore, it can be said that the ratio σ_(max)/σ_(n) between the mean stress σ_(n) and maximum stress σ_(max) isprecisely equal to the factor of stress concentration α. Meanwhile, inthe stress region where the echo height h is nearly constant, themaximum stress σ_(max) in the stress-concentrated portion exceeds theyield point σ_(y) of the test piece material and the strain increases inthe condition without any change in stress of the stress-concentratedportion, which is caused by the so-called "sliding deformation". In thestress region in which the echo height h increases, the test pieceincurs a further increased plastical deformation, which, it isestimated, is caused by the change in shape of the stress-concentratedportion in any manner other than the elastic deformation.

Also as apparent from FIG. 8, when the mean stress σ_(n) is a maximum ofabout 10 kg/mm², the echo height h has a larger gradient as the radiusof curvature σ is smaller. On the other hand, it was reported on thepage 632 of the paper entitled "Stress Concentration" (published bbyMorikita Shuppan in 1973) that the solid solution of factor of stressconcentration was obtained in the following form as the results ofexperiments on the photo-elasticity: ##EQU6##

By calculating the factor of stress concentration α from the expression(12) and determining the constant of proportion a in the above-mentionedexpression (10), by the method of least squares, as to each radius ofcurvature ρ in the experiment results shown in FIG. 8, the relationbetween the factor of stress concentration and the constant ofproportion is plotted as indicated with small circles in FIG. 9. As seenfrom FIG. 9, there is a linear relation between the logarithmic value ofthe factor of stress concentration α and the constant of proportion aindicative of the gradient h/σ_(n). The regression expression of theconstant of proportion a determined, by the method of least squares, inrelation with the factor of stress concentration α will be as follows:

    a=-0.158 log α                                       (13)

It is seen that the above expression depicted with a straight linegenerally coincides with the plotting with small circles as shown inFIG. 9. The error of the straight linn from each plotted point is assmall as less than 20% in case the factor of stress concentration α isless than 5. As seen evident from the relation between the mean stressσ_(n) and echo height h, the factor of stress concentration α could bedetermined with an accuracy sufficiently high in practice andquantitatively. The relation between the mean stress σ_(n) and echoheight h is expressed as follows based on the expressions (10) and (13)

    h=(-0.158 log α)σ.sub.n                        ( 14)

(2) It is considered from the expression (12) that the factor of stressconcentration α of the test piece used in the above-mentioned experimentwas influenced by the radius of curvature ρ as well as the test piecewidths B and b. Experiment was effected in the procedure shown in FIG. 6as in FIG. 8 (the probe of 5 MHz was used) on the test pieces of thesame shape as in FIG. 4 and radii of curvature ρ: 0.74. 0/76, 1.65 and3.50 mm but with the small width b of 29 mm instead of 20 mm. Theresults of this experiment are shown in FIG. 10. As shown in FIG. 10,the relation between the mean stress σ_(n) and echo height h is similarin numerical value and gradient to the results shown in FIG. 8. Therelation between the constant of proportion a indicative of the gradientin the region where the expression (10) is established (mean stressσ_(n) being on the order of 10 to 15 kg/cm²) and the factor of stressconcentration α can be determined from the expression (12), and isindicated with small circles in FIG. 11. The regression expression ofthe constant of proportion a in relation with the factor of stressconcentration α was determined to be identical to the expression (13),and it is shown with a straight line in FIG. 11.

(3) The results of an experiment conducted on the influence of therefraction angle of the probe on the relation between the mean stressσ_(n) and echo height h is shown in FIG. 12. The parameter is therefraction angle of the probe. The test piece used in the experiment hasthe same shape as in FIG. 4 and is the same in other respects as in FIG.4 except for the radius of curvature ρ being 0.66 mm. Also, the probesused are of 4 MHz in frequency as in the above experiment and of threekinds of refraction angle: 45°, 60° and 70°. The factor of stressconcentration α of this test piece is 3.43 as calculated based on theexpression (12). As known from the results of this experiment, therelation between the mean stress σ_(n) and echo height h is so littleinfluenced by any difference in refraction angle among the probes thatit may be depicted with a straight line. However, use of a probe largerin refraction angle, which permits to effect an ultrasonic measurementfrom a place distant from the stress-concentrated portion so that thedeformation of the probe-attached portion due to the stress in thestress-concentrated portion is minimized, improves the accuracy ofmeasurement.

(4) FIGS. 13 and 14 show the results of the experiments conducted on howthe change of the probe frequency f influenced on the relation betweenthe mean stress σ_(n) and echo height h. The test piece used in theexperiment has the same shape as in FIG. 4 and it is the same in otherrespects as in the above-mentioned experiment except for the radii ofcurvature ρ of the fillet weld being 0.08, 0.21, 0.39, 0.66 and 2.75 mm(5 kinds). The probes used in this experiment include a one of 2 MHz infrequency (2Z10×10A70) with the results in FIG. 13 and another of 7 MHzin frequency (7Z10×10A70) with the results in FIG. 14. So, theexperiments were done in the same manner as having been described withreference to FIG. 6 as in FIG. 8 (the probe of 5 MHz was ued). Both theexperiment results in FIGS. 13 and 14 have a same tendency as that inFIG. 8 which has been described in (1) above. From these experimentresults, the following was proved. Namely, (i) The higher the probefrequency f, the greater the gradient of the echo height h, namely, thevalue of the constant of proportion a in the expression (10), so thatthe accuracy of measurement is so higher. (ii) The lower the frequencyf, the smaller the variation of the echo height h with respect to themean stress σ_(n) and the higher the linearity of the gradient. Asapparent from these experiment results, a probe frequency on the orderof 5 MHz meet the requirements for both the accuracy of measurement andthe linearity of gradient. As having been described in the above, therefraction angle of the probe should preferably be large. Thus, theprobe which will be referred to in the following explanation is5Z10×10A70 of 70° in refraction angle unless otherwise noted.

Experimental expressions on the relation between the mean stress σ_(n)and echo height h determined with probe frequencies f of 2 MHz and 7 MHzby the method described in (1) will be as follows:

With 2 MHz:

    h=(-0.07 log α)σ.sub.n                         ( 15)

With 7 MHz:

    h=(-0.185 log α)σ.sub.n                        ( 16)

The relations between the gradients in the experimenral expressions(14), (15) and (16) and each of the frequencies f are shown with smallcircles in FIG. 15. The horizontal axis of the graph in FIG. 15 is thelogarithmic value of frequency f (MHz) while the vertical axis is thegradient A of the experimental expression. As seen from FIG. 15, thereis a linear relation between the gradient A and the logarithmic value offrequency f. The regression expression of this relation is as follows:

    A=-0.213 log f-0.007                                       (17)

where f: Probe frequency (MHz)

The relation between the mean stress σ_(n) and echo height h, includingthe probe frequency f, can be derived as follows from the expression(17):

    h=[(-0.213 log f-0.007) log α]σ.sub.n          ( 18)

The echo height h can be expressed in the form of a simple expressionwhich can be determined from a probe frequency f, factor of stressconcentration α and mean stress σ_(n). The factor of stressconcentration α can be expressed as follows based on the expression(18): ##EQU7##

The relation (19) can be used to determine a factor of stressconcentration α from the values of an echo height h, probe frequency fand mean stress σ_(n). Among these factors, the frequency f is a knownvalue depending upon a probe to be used, and the value of the meanstress σ_(n) can be easily determined from a known sectional area of amember under measurement if the value of a load working on the memberhaving a stress-concentrated portion is known. In effect, by determiningthe value of the third factor, namely, the echo height h, on the CRTdisplay of the measuring apparatus, the factor of stress concentration αcan be determined from the expression (19).

Therefore, with the present invention, it is possible to determine thefactor of stress concentration in the stress-concentrated portion of ato-be-measured member on the basis of the expression (19) easily,quantitatively, highly accurately and nondestructively. Also, the factorof stress concentration can be measured real-time by measuring theheight of an echo from the stress-concentrated portion of the member ofan actual product. As having been described in the foregoing, the factorof stress concentration can be determined directly, in a short time andwith an accuracy equal to or higher than the value determined in thephoto-elasticity experiments, by measuring the height of an echo fromthe stress-concentrated portion of a member under measurement. At thesame time, the factor of stress concentration can be easily determinedstatically as well as dynamically by changing the working stress.

These and other objects and advantages of the present invention will bebetter understood from the ensuing description made, by way of example,of the preferred embodiments of the present invention with reference tothe drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 explains the basic principle of the inventive method of measuringthe factor of stress concentration;

FIG. 2 is a detail view of the portion a of the stress-concentratedportion in FIG. 1, showing the stressed state of σ_(n1) in mean stress;

FIG. 3 explains the change from the stressed state in FIG. 2 to astressed state of σ_(n2) in mean stress;

FIG. 4 explains the shape of the test piece used in the experimenteffected to prove the basic principle of the inventive method ofmeasuring the factor of stress concentration;

FIG. 5a shows the fillet weld zone of a Tee joint, and FIG. 5b shows themodeling of the fillet welded portion in FIG. 5a by finishing the filletweld zone so as to have a radius of curvature ρ;

FIG. 6 explains the experimental device using the test piece in FIG. 4;

FIG. 7 explains as enlarged in scale the display on the CRT screen ofthe experimental device in FIG. 6;

FIG. 8 shows one example of the result of an experiment in which therelation between the mean stress and echo height was determined takingas parameter the radius of curvature of the stress-concentrated portion;

FIG. 9 shows the correlation between the factor of stress concentrationin FIG. 8 and the constant of proportion indicative of the gradient ofthe relation between the mean stress and echo height;

FIG. 10 shows one example of the result of an experiment in which therelation between the mean stress and echo height was determined with thetest piece width and radius of curvature of the stress-concentratedportion changed;

FIG. 11 shows the correlation between the factor of stress concentrationin FIG. 10 and the constant of proportion indicative of the gradient ofthe relation between the mean stress and echo height;

FIG. 12 shows one example of the result of an experiment in which therelation between the mean stress and echo height was determined takingas parameter the angle of refraction of the ultrasound probe;

FIG. 13 shows one example of the result of the experiment in FIG. 8having been done with the probe frequency changed;

FIG. 14 shows the result of the experiment in FIG. 13 having been donewith the probe frequency changed;

FIG. 15 shows the correlation between the gradient of the experimentalexpression by which the relation between the mean stress and echo heightwas determined with the probe frequency changed, and that probefrequency;

FIGS. 16a to 18 show a first example in which the present invention wasapplied to the measurement of the factor of stress concentration at theweb side of fillet weld zone of a Tee joint, FIG. 16a explaining thetest piece and measuring apparatus, FIG. 16b being a side elevation ofthe test piece in FIG. 16c showing as enlarged in scale the toe of weldW of the portion to be measured, FIG. 17 showing the result of themeasurement of the relation between the mean stress and echo heighttaking as parameter the radius of curvature of the toe of weld W, andFIG. 18 being a graph indicative of the relation between the radius ofcurvature and factor of stress concentration, determined from themeasurement result in FIG. 17, in which the regression line of the aboverelation is indicated with a dot line;

FIGS. 19a to 21 show a second example in which the present invention wasapplied to the measurement of the factor of stress concentration at theflange side of fillet weld zone of a Tee joint, FIG. 19a explaining thetest piece and measuring apparatus, FIG. 19b being a side elevation ofthe test piece in FIG. 19a, FIG. 19c showing as enlarged in scale thetoe of weld E of the portion to be measured, FIG. 20 showing the resultof the measurement of the relation between the mean stress and echoheight taking as parameter the radius of curvature of the toe of weld E,and FIG. 21 being a graph indicative of the relation between the radiusof curvature and factor of stress concentration, determined from themeasurement result in FIG. 20, in which regression line of the aboverelation is indicated with a dot line;

FIGS. 22a to 24 show a third example application of the presentinvention, FIG. 22a showing the shape of an undercut developed in thetoe of weld of a test piece, FIG. 22b showing an example shape of theundercut as enlarged in scale, FIG. 23 showing the result of measurementof the relation between the mean stress and echo height taking asparameter the ratio between the radius of curvature and depth of theundercut bottom, and FIG. 24 being a graph indicative of the relationbetween the ratio between the radius of curvature and depth and thefactor of stress concentration, determined from the measurement resultin FIG. 23, in which the relation calculated from the experimentalexpression in an experiment of photo-elasticity is indicated with a dotline;

FIGS. 25 and 26 explain a fourth example application of the presentinvention, FIG. 25 showing the measurement result of the relationbetween the mean stress and echo height taking as parameter the ratiobetween the radius of curvature and depth of the bottom of an undercutdeveloped in the toe of weld at the flange side, and FIG. 26 being agraph indicative of the relation between the ratio between the radius ofcurvature and depth and the factor of stress concentration, determinedfrom the measurement result in FIG. 25, in which the relation calculatedfrom the experimental expression in an experiment of photo-elasticity.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

The present invention will be further explained with reference to thedrawings, taking as model the fillet weld zone of a Tee joint.

FIGS. 16 to 18 explain a first embodiment of the present invention,among which FIGS. 16a and 16b provide explanation of the test piece andan apparatus to measure the factor of stress concentration in the weldzone of the test piece. The test piece has the flange width B of 40 mmat the T joint and web width b of 8 mm, the flange thickness l₁ and webheight l₂ both of 200 mm, and the length t of 50 mm. The weld zone is aweld bead of 8 mm leg length, which is formed by a semi-automatic CO₂arc welding. FIG. 16c shows as enlarged in scale a toe of weld W at theweb side which is the stress-concentrated portion in FIG. 16a. The toeof weld W has 5 kinds of radius of curvature ρ_(w) : 0.4, 1.7, 1.9, 4.2and 6.2 mm. The radii of curvature ρ_(w) were measured in the samemanner as having been described concerning the test piece shown in FIG.4. The material of the test piece is SM50A, the same as that of the testpiece shown in FIG. 4. The toes of weld other than that W of the memberunder measurement were finished by a grinder so as to have asufficiently large radius of curvature to prevent the toe of weld W frombeing influenced by any stress concentration. The measuring apparatusfor factor of stress concentration is similar to that shown in FIG. 6.The probe 3 placed on the web is secured by any adhesive for theultrasound to be omitted for incidence toward the toe of weld W at theweb where stress is concentrated, and then electrically connected to theultrasound flaw detector 7. The test piece is mouned in place on theAmsler universal testering machine (not shown) and applied with atensile load F in the direction of arrow with the load being changed invalue. The types of the ultrasound flaw detectors 7 are SM-80 and SM-90by Tokyo Keiki Co., Ltd. The type of the probe 3 is 5Z10×10A70 of whichthe frequency is 5 MHz and refraction angle is 70°.

When the test piece is applied with a variety of tensile loads F, a meanstress σ_(nw) develops against each of the tensile loads F in the web.In the stressed state in which mean stresses σ_(nw) of different valuesdevelop, the reflected wave derived from the incident wave 4 emittedfrom the probe for incidence toward the toe of weld W at the web sidehas the beam path length displayed along the horizontal axis of CRT 8,while height h of the echo 10 indicative in dB of the acoustic pressureof the reflected wave from the toe of weld B is displayed along thevertical axis.

The measurement results of the relation of the echo height h with thechange of the mean stress σ_(nw) are shown in FIG. 17. The parameter isthe radius of curvature ρ_(w) of the toe of weld W at the web side. Asseen from FIG. 17, the smaller the radius of curvature ρ_(w), thegreater the change of echo height h with the change of mean stressσ_(nw) and so the gradient of the echo height in the range of about 10to 15 kg/mm² in mean stress σ_(nw) in which the relation between boththese factors is nearly linear, is great as compared with that with alarger radius of curvature ρ_(w). By substituting this relation in theexpression (19), a factor of stress concentration α of the toe of weld Wcan be easily determined for each mean stress σ_(nw). The relationbetween the radius of curvature ρ_(w) and the factor of stressconcentration α obtained from the expression (19) is shown with smallcircles in FIG. 18. The regression line, formed by connecting the smallcircles, which indicates the relation between the factor of stressconcentration α and radius of curvature ρ_(w), is expressed as follows:##EQU8## As seen from this relation, the factor of stress concentrationα can have an approximate value determined from only the relation withthe radius of curvature ρ_(w). As also proved with this firstembodiment, the plotting with dots of the factor of stress concentrationα determined by the expression (20) coincides relatively well with thatwith small circles of the factor of stress concentration α determined bythe expression (19), and the expression (20) affectable with only theradius of curvature irrespective of any leg length, shape, thickness,etc. of the weld is practically usable by determining the radius ofcurvaure.

FIGS. 19a to 21 explain a second embodiment of the inventive method ofmeasuring the factor of stress concentration. According to the secondembodiment, the factor of stress concentration of the toe of weld E atthe flange side is measured, while the first embodiment is intended formeasuring the factor of stress concentration of the toe of weld W at theweb side.

FIGS. 19a and 19b explain the test piece and the apparatus formeasurement of the factor of stress concentration in the weld, and FIG.19c shows as enlarged in scale the toe of weld E at the flange side inFIG. 19a. The test piece is the same in shape, material and dimensionsas in the first embodiment, except for the variety of radius ofcurvatures ρ_(E) being in 4 kinds: 0.4, 1.34, 3.7 and 6.0 mm. Any toesof weld other than that E of the test piece are finished by a grinder soas to have a sufficiently large radius of curvature to prevent the toeof weld E from being affected by any stress concentration. Also, themeasuring apparatus for factor of stress concentration and the method ofmeasurement are identical to those in the first embodiment, providedthat the probe 3 is placed at the flange side for the ultrasound wave 4to be emitted from the probe for incidence toward the toe of weld E atthe flange side.

The relation of the echo height h with the change of mean stress σ_(nw)was measured and the result is shown in FIG. 20. The parameter is theradius of curvature ρ_(E) of the toe of weld E at the flange side. Asseen from FIG. 20, the smaller the radius of curvature ρ_(E), thegreater the change in echo height h as the mean stress σ_(nw), which arethe same as in FIG. 17 of the first embodiment; however, with the radiiof curvature ρ_(E) being 3.7 and 6.0 mm, the echo height h shows only alittle change in the range of mean stress σ_(nw) being a maximum of 15kg/mm². This means that the factor of stress concentration α is small.The relation between the radius of curvature ρ_(w) and the factor ofstress concentration α determined from the expression (19) is shown withsmall circles in FIG. 21. The regression line formed by connecting thespots plotted with the small circles to each other and indicative of therelation between the factor of stress concentration α and radius ofcurvature ρ_(E) can be expressed as follows: ##EQU9## It is seen fromthis expression that the factor of stress concentration α can beapproximated from only the relation with the radius of curvature ρ_(E).Plotting in FIG. 21 with dot line of the factors of stress concentrationα determined by the expression (21) is relatively well coincident withthe small circles indicative of the factors of stress concentration αdetermined by the expression (19). With the radius of curvature ρ_(E)increased up to some 7.0 mm, the factor of stress concentration αbecomes less than 1.2. On the contrary, with a small radius of curvatureρ_(E) being as small as 0.4 mm, the factor of stress concentration αbecomes 1.79. This value is small as compared with the values of 1.38and 2.58 as shown in FIG. 18 of the first embodiment. It means that evenwith a slight difference in measuring position between the toe of weld Wat the web side and that E at the flange side while the measurement isbeing done under same conditions, a difference in value of the factor ofstress concentration can be definitely measured, which proves one aspectof the present invention that the measurement of the factor of stressconcentration can be done with a high accuracy.

A third embodiment of the inventive method of the factor of stressconcentration will be explained with reference to FIGS. 22a to 24.According to this third embodiment, an undercut W_(u) is developed inthe toe of weld, in place of the toe of weld W at the web side in thefirst embodiment, during a normal welding, and the factor of stressconcentration at the bottom of the undercut W_(u) is measured.

The test piece, measuring apparatus for the factor of stressconcentration and method of measuring according to this third embodimentare identical to those in the first embodiment, except that the toe ofweld W in FIG. 16a, having been described concerning the firstembodiment, is replaced with the undercut shown in FIG. 22a. The radiusof curvature ρ_(wu) of the undercut bottom and the shape of the undercutof a bottom d from the plate surface to the bottom are shown by way ofexample in FIG. 22b. This FIG. 22b shows as enlarged in scale by 50times the example radius of curvature ρ_(wu) of 0.07 mm and depth d of0.63 mm. The reference symbol D denotes a weld beam surface of which thetoe of weld other than the undercut is finished sufficiently to preventthe toe of weld w_(u) from being affected by any stress concentration.ρ_(wu) /d in this embodiment is 0.11. The radius of curvature ρ_(wu) andthe depth d were measured by modeling the undercut by a replica, cuttingthe model into slices, and magnifying the slice by a light projector formeasurement. FIG. 22a is a schematic drawing of FIG. 22b. While in thefirst and second embodiments, several kinds of radius of curvature ofthe toe of weld were measured, the ratio between the radius of curvatureρ_(wu) and depth of d of the undercut bottom, namely, ρ_(wu) /d, ismeasured and the factors of stress concentration are measured of testpieces of three kinds of that ratio: 0.11, 0.14 and 0.67.

The results of measurement of the relation of the echo height h with thechange of mean stress ρ_(nw) are shown in FIG. 23. The parameter isρ_(wu) /d. As apparent from FIG. 23, the smaller the value of the ratioρ_(wu) /d, namely, the smaller and sharper the radius of curvatureρ_(wu) of the undercut bottom and the larger the depth d, the steeperthe gradient of the echo height h. By substituting this relation in theexpression (19), the factor of stress concentration α at the undercutW_(u) developed in the toe of weld at the web side can be easilydetermined.

FIG. 24 shows the relation between the factor of stress concentration αdetermined by the expression (19) and the ratios ρ_(wu) /d of said threekinds of test piece. The points of the factors of stress concentrationare indicated with small circles in this FIG. 24. Concerning a testpiece with a U-shaped nocth, the factor of stress concentration when atensile load works in the direction in which the opening of the notch isopened had been measured in the past in a variety of photo-elasticityexperiments, and experimental expressions obtained by the experimentshad been proposed. For example, H. Neuber reported a followingexpression in 1958: ##EQU10## E. Inglis reported an expression asfollows in 1913: ##EQU11## Also E. Armbruster reported the followingexpression in 1931: ##EQU12## Plotting with a dot line in FIG. 24 of therelation between the factor of stress concentration α determined by theexpression (22) among those (22) to (24) and the ratio ρ/d is relativelywell coincident with the small circles indicative of the relationbetween the factor of stress concentration α determined by theexpression (19) and the ratio ρ_(wu) /d of the three kinds of testpiece, especially at the side where the value of the factor of stressconcentration α is large. The expression (22) also proves that with theinventive method, the factor of of stress concentration at the bottom ofany undercut developed during a welding can be determined as hardlyinfluenced by the bead shape, leg length, thickness, etc. of the weldzone, if the value of ρ_(wu) /d can be measured, with such a highaccuracy that the inventive method can be satisfactorily used inpractice.

A fourth embodiment of the inventive method will be explained withreference to FIGS. 25 and 26. While the third embodiment is intended formeasurement of the factor of stress concentration at the bottom of anundercut W_(u) developed in the toe of weld at the web side, the fourthembodiment is to measure the factor of stress concentration at thebottom of an undercut E_(u) developed in the toe of weld at the flangeside.

The test piece, measuring apparatus for the factor of stressconcentration, method of measuring, etc. in the fourth embodiment areidentical to those in the second embodiment, except that the toe of weldE in FIG. 19a, having been described concerning the second embodiment,is replaced with the undercut shown in FIG. 22a. In this embodiment, therelations between the radius of curvature ρ_(Eu) of the undercut bottomand the depth d from the plate surface to the undercut bottom, namely,ρ_(Eu) /d, are in thress kinds: 0.04, 0.32 and 0.80. The radius ofcurvature ρ_(Eu) and the bottom d of the test piece were measured in thesame manner as in the third embodiment.

The results of the measurement of the relation between the echo height hwith the change of mean stress σ_(nw) are shown in FIG. 25. Theparameter is ρ_(Eu) /d. The decrease of the echo height h as the meanstress σ_(nw) increases, it is apparent from FIG. 25, is very dull ascompared with that in FIG. 23 of the third embodiment, and also ascompared with the factor of stress concentration of an undercutdeveloped in the toe of weld at the web side, that of the undercut inthis fourth embodiment is very small. By substituting this relation inthe expression (19), the factor of stress concentration α at the bottomof the undercut E_(u) with respect to the mean stress σ_(nw) can beeasily determined.

FIG. 26 shows the relation between the factor of stress concentration αdetermined by the expression (19) and the ratios ρ_(Eu) /d of the threekinds of test piece, with the relations being plotted with smallcircles. As having been described in the description concerning thethird embodiment, many photo-elasticity experiments for determing thefactor of stress concentration had been done on test pieces withU-shaped notch; as the results, the expressions (22) to (24) had beenreported. Plotting with dot line in FIG. 26 of the relation between thefactor of stress concentration α determined by dividing by 1.5 thesquare root of the expression (22) reported by H. Neuber, namely, by theexpression (25), and the ratio ρ/d is nearly coincident with the smallcircles determined by the expression (19): ##EQU13## Therefore,similarly to the third embodiment, the inventive method is proved topermit to measure the factor of stress concentration at the bottom of anundercut developed in the toe of weld at the flange side without beingaffected by the bead shape, leg length, thickness, etc. of the weld zonebut with such a high accuracy that the inventive method can besatisfactorily used in practice, if the value of ρ_(Eu) / can bemeasured.

The method having been described in the foregoing is a visual method bywhich an echo is displayed on the CRT for measurement, but according tothe present invention, the analogue quantity of echo height can bedigitized by a well-known converting means and the correlation betweenthe analogue quantity of the echo height and the working stress becalculated, for thereby indicating the results as numerical values.Also, such values can be stored in a memory and compared with referencevalues as necessary for diagnosis of the fatigue of the members of aproduct or machine or for preventive inspection of the latter.

It is evident to those skilled in the art that the present invention isnot limited to the embodiments having been described in the foregoing,but can be varied in various forms without departing from the scope oftechnical concept of the present invention.

What is claimed is:
 1. A method of measuring the stress concentrationfactor at a stress-concentrated portion of a mechanical member having anundercut located at a portion where the cross-section of the membervaries abruptly, said undercut extending into the member and having abottom, said method comprising:causing an ultrasonic wave to impinge onsaid stress-concentrated portion of said member in a first stressedstate; detecting the acoustic pressure of the ultrasonic wave reflectedfrom the bottom of the undercut while said member is in said firststressed state; gradually increasing or decreasing the stress on saidmember to cause an increased or decreased stressed state and in adirection for causing a variation of the shape of the bottom of theundercut; continuing to cause said ultrasonic wave to impinge on saidstress-concentrated portion of said member; detecting the acousticpressure of the ultrasonic wave reflected from said bottom of theundercut while said undercut is in the increased or decreased stressedstate; comparing the acoustic pressure of the ultrasonic wave reflectedfrom the bottom of the undercut in said first stressed state with theacoustic pressure when said stress concentrated portion is in theincreased or decreased stressed state; and determining the stressconcentration factor at the stress-concentrated portion using as anevaluation index the change ratio of the compared values of the acousticpressure.
 2. The method as set forth in claim 1 in which said step ofincreasing or decreasing the stress is effected within the limits of theelasticity of said member.
 3. The method as set forth in claim 2 inwhich said step of increasing or decreasing the stress is effectedwithin a range of stress in which as said stress increases, the detectedacoustic pressure of the ultrasonic wave reflected from thestress-concentrated portion decreases.
 4. The method as set forth inclaim 1 in which said step of determining the stress concentrationfactor comprises using as the evaluation index the value of thecorrelation between the change ratio of the comparison values and thestress acting on said stress-concentrated portion.
 5. The method as setforth in claim 1 in which said step of causing an ultrasonic wave toimpinge on said stress-concentrated portion comprises the steps ofplacing a probe in close contact with a surface of the member having astress-concentrated portion in a stressed state, and impinging saidultrasonic wave on said stress-concentrated portion with the probe keptin place at the point of close contact.
 6. The method as set forth inclaim 5 in which said step of placing the probe comprises placing it ata position spaced from the stress-concentrated portion, which positionis not influenced by the stress in said stress-concentrated portion.